12 1. PRELIMINARIES

where

gradf(x) =

∂f

∂xji

n

i,j=1

.

Thus the standard Poisson-Lie bracket becomes

(1.25)

{f1,f2}SLn (x) =

1

2

(R(gradf1(x) x), gradf2(x) x)−

1

2

(R(x gradf1(x)),x gradf2(x)),

where the action of the R-matrix (1.23) on ξ = (ξij)i,j=1 n ∈ sln is given by

R(ξ) = (sign(j − i)ξij)i,j=1

n

.

Substituting into (1.25) coordinate functions xij, xkl, we obtain

{xij,xkl}SLn =

1

2

(sign(k − i) + sign(l − j)) xilxkj.

In particular, the standard Poisson-Lie structure on

SL2 =

a b

c d

: ad − bc = 1

is described by the relations

{a, b}SL2 =

1

2

ab, {a, c}SL2 =

1

2

ac, {a, d}SL2 = bc,

{c, d}SL2 =

1

2

cd, {b, d}SL2 =

1

2

bd, {b, c}SL2 = 0.

Note that the Poisson bracket induced by {·, ·}SL2 on upper and lower Borel

subgroups of SL2

B+ =

p q

0

p−1

, B− =

p 0

q

p−1

has an especially simple form:

{p, q} =

1

2

pq.

The example of SL2 is instrumental in an alternative characterization of the

standard Poisson-Lie structure on G. Fix i ∈ [1,l], l = rank g, and consider an

annihilator of the subalgebra

gi

= span{ei,e−i,hi} with respect to the Killing

form:

(gi)⊥

= hi

⊥

⊕

(

⊕α∈Φ\{α±i}gα

)

,

where hi

⊥

is the orthogonal complement of hi in h with respect to the restriction

of the Killing form. Though

(gi)⊥

is not even a subalgebra in g, it is an ideal in

g∗

equipped with the Lie bracket (1.24). It follows that for every i, the image of

the embedding ρi : SL2 → G defined in (1.7) is a Poisson-Lie subgroup of G. The

restriction {·, ·}ρi(SL2) of the Sklyanin bracket (1.22), (1.23) to ρi(SL2) involves

only the restriction of the standard R-matrix to

gi

= ρi(sl2). It is not diﬃcult to

check then, that

{·, ·}ρi(SL2) = (αi,αi){·, ·}SL2 .

Thus the standard Poisson-Lie structure is characterized by the condition that for

every i ∈ [1,l], the map

(1.26) ρi :

(

SL2, (αi,αi){·, ·}SL2

)

→

(

G, {·, ·}G

)

is Poisson.